Spectrality of product-form self-similar measures and tiles

Abstract

This paper studies the Fourier properties of self-similar measures and tiles generated by digit sets of product-form. Let $0<\rho <1$ be a real number and let $D$ be the direct sum of two consecutive integer sets: $$D=\{0,1,\cdots ,N-1\}\oplus m\{0,1,\cdots ,L-1\},$$ where $N,m,L\in {\mathbb{N}}^{\ast }$ with %$N,L\ge 2$ $N,L\ge 2$. The pair $(\rho ,D)$ determines the self-similar iterated function system (IFS) $\{{\varphi }_d(\cdot )=\rho (\cdot +d){\}}_{d\in D}$. Let ${\mu }_{\rho ,D}$ and $T$ be the associated self-similar measure and self-similar set, respectively. We first prove that $L^2({\mu }_{\rho ,D})$ admits an exponential orthonormal basis if and only if ${\rho }^{-1}=p\in \mathbb{N}$ satisfies $N\mid p$, $L\mid p$ and $N\mid \frac{m}{\gcd (m,p^d)}$, where $$d=\max \left\{i:\gcd \left(\frac{mL}{\gcd (mL,p^i)},L\right)\ne 1,i\in \mathbb{N}\right\}.$$ This result extends a series of previous studies, including the cases where $N,L$ are primes [An-Wang, J. Funct. Anal., 2021] and $N=L$ [Liu-Peng-Wu, J. Math. Anal. Appl., 2019]. Furthermore, in the context of the Fuglede conjecture, we show that when ${\rho }^{-1}=\#D=NL$, the space $L^2({\chi }_{T}dx)$ admits an exponential orthonormal basis if and only if $T$ is a translation tile of $\mathbb{R}$.